Electrons secular equation

The method of many-electron Sturmian basis functions is applied to molecnles. The basis potential is chosen to be the attractive Conlomb potential of the nnclei in the molecnle. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nnclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

From the one-electron secular equation, it follows that the gerade solution (in momentum space) has the form ... 

Koga and others [10] have shown that the loss of accuracy resulting from tmncation of the basis set can be reduced by replacing the one-electron secular equation, (34), with an equation based on a second iteration of the integral equation (31). The second-iterated integral equation has the form ... 

Molecular Orbitals Based on Sturmians 5.1 The One-Electron Secular Equation... 

The variational problem may again be formulated as a secular equation, where the coordinate axes are many-electron functions (Slater determinants), <, which are orthogonal (Section 4.2). 

There seems to be a certain complementarity between the degree of difficulty in evaluating HKL for various one-electron sets y)k and the order of the secular equation needed to obtain a certain accuracy in the result. The work carried out in getting extensive tables of molecular integrals has also been of essential value for facilitating the calculation of the matrix elements HKL. 

Table X gives an idea of the strength of the various expansion methods, and it shows that, by using the principal term only, one can hardly expect to reach even the above-mentioned chemical margin, even if the wave function W gO(D) is actually very close in the helium case. This means that one has to rely on expansions in complete sets, and the construction of the modern electronic computers has fortunately greatly facilitated the numerical solution of secular equations of high order and the calculation of the matrix elements involved. For atoms, the development will probably go very fast, but, for small molecules one has first to program the conventional Hartree-Fock scheme in a fully self-consistent way for the computers, before the next step can be taken. For large molecules and crystals, the entire situation is much more complicated, and it will hence probably take a rather long time before one can hope to get a detailed understanding of the correlation phenomena in these systems.

Diagonalizing the corresponding 2x2 secular equation and some algebraic manipulation gives the four-electron ground-state potential surface... 

For each value of j there is a set of equations, but the different sets are all identical except for the replacement of the subscript j by some other, say j. The condition for the solubility of this set of equations is that their determinants vanish. The l roots of this secular equation, W = Wj (j = 1,2,...,/) give the energies of the l molecular orbitals. We then assign two electrons (one with positive and one with negative spin) to the lowest... 

The diagrams are best understood in terms of the apparent repulsion between the energy levels of combining systems, which can easily be related to a perturbation treatment of the secular equations. For example, two carbon atom ir electron levels (1) and (2) with energies ao would interact to remove the degeneracy... 

The change x may represent the effect of a substituent or heteroatom at the wth position. In this case the v electron energy levels, charges g free valences Fg and bond orders pgt can be obtained by direct solution of the secular equations (8) using... 

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